Operator Representation and Logistic Extension of Elementary Cellular Automata

Operator representation and logistic extension of elementary cellular automata

M. Ibrahimia, A. Guc¸l¨ u¨b,c, N. Jahangirovb, M. Yamand, O. Gulseren¨ b,e, S. Jahangirovb,f

aCentre de Physique Th´eorique(CPT), Turing Center for Living Systems, Aix Marseille Universit´e,13009 Marseille, France bUNAM-Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey cDepartment of Electrical and Electronics Engineering, Bilkent University, 06800, Ankara, Turkey dDepartment of Aeronautical Engineering, University of Turkish Aeronautical Association, 06790, Ankara, Turkey eDepartment of Physics, Bilkent University, Ankara 06800, Turkey fInterdisciplinary Graduate Program in Neuroscience, Bilkent University, 06800, Ankara, Turkey

Abstract We redeﬁne the transition function of elementary cellular automata (ECA) in terms of discrete operators. The operator representation provides a clear hint about the way systems behave both at the local and the global scale. We show that mirror and complementary symmetric rules are connected to each other via simple operator transformations. It is possible to decouple the representation into two pairs of operators which are used to construct a periodic table of ECA that maps all unique rules in such a way that rules having similar behavior are clustered together. Finally, the operator representation is used to implement a generalized logistic extension to ECA. Here a single tuning parameter scales the pace with which operators iterate the rules. We show that, as this parameter is tuned, many rules of ECA undergo multiple phase transitions between periodic, locally chaotic, chaotic and complex (Class 4) behavior.

Emergence, a semantic gap between behavior and interac- is a simple, yet often efficient approach for predicting the class tions, is a hallmark of dynamical systems. Examples of emer- [8]. Also, further approaches have introduced entropies, mean gent behavior include: exchanging deals between thousands field descriptions or network analyses that help to understand of agents/companies making up the whole stock market; inter- the relation between patterns and their respective rules [9, 10, actions between alternately expressing genes generating juxta- 11, 12, 13]. However, these analyses are either too generic to posed biological forms; series of firings between peculiarly in- hold for, or too specific to apply to larger families of CA. Con- terconnected neurons breeding functional activities in the brain. sequently, the quest for generalizing a method to any dynamical Given the particularities of such systems (multiple levels of lattice system remains challenging and this motivates the need hierarchies, events and processes operating at broadly differ- to approach CA rules in the light of a different perspective. ent space/time scales), complexity science has adopted cellu- Using the ECA set as an example, we suggest a fundamen- lar automata (CA) [1] as much simpler computational models tal approach that redefines the transition function based on a to specifically target the semantic gap between individual and simple intuition gained by visual inspection of the system scale global degrees [2, 3, 4]. These agent based models operate in dynamics. This approach brings the microscopic information fully discrete domains and are known to generate large scale closer to the large scale dynamics and thus helps understand types of behavior only through local interactions (rules) [5, 6]. the properties of a system based on its “first principles”. While aiming to acquire a generic understanding applicable Having distilled the interactions into iterative operations, to all dynamical systems, a main hypothesis has gathered sev- we employ these operations and the symmetries of the system to eral attempts to bridge the spatio-temporal patterns observed rewrite the transition function in a more intuitive notation. Our in CA (phenotype) with their rule space (genotype). The lead- approach provides a framework that links the similarities and arXiv:2010.02073v1 [nlin.CG] 5 Oct 2020 ing apprehension is Wolfram’s classification which postulates differences observed at the phenotype level to an operator based that the asymptotic behavior of a dynamical system lies in one translation of the rule space. Furthermore, this framework en- of these four classes: homogeneous, periodic, aperiodic and ables us to implement a generalized logistic extension to ECA complex behavior [2, 7]. He introduced elementary cellular [14] where a single parameter scales the pace with which op- automata (ECA) as a paradigmatic simple set of rules which erators iterate the system. As a result, the binary state space of comprise all these types. ECA is expanded into a Cantor set and in turn we get a chance to Several studies have attempted to understand how distinct observe transitions between classes [5]. In particular, we reveal groups of rules act to generate similar types of asymptotic be- several complex (Class 4) instances that are not reachable in havior, eventually making up the four classes. Langton’s method the standard ECA. More interestingly, the behavioral difference of labelling a parameter out of a unit interval to a certain rule between some rules sharing similar genetic code is diminished upon the logistic extension. ECA are time dependent one dimensional infinite strings Email address: [email protected] (S. Jahangirov)

Preprint submitted to Elsevier October 6, 2020 t t ∞ of sites S = {sn}|n=−∞ of a binary state space si ∈ {0, 1}. In ECA a Rule defines how the value of a certain site is iterated t+1 t si = fS si based on its current value and the values of its t t t nearest neighbors, through a transition function fS (si−1, si, si+1). Given the binary state space, there are eight possible configurations of a three site neighborhood, resulting in 28 = 256 possible mappings, i.e rules. Mapping of Rule 30 is shown as an example in Fig. 1(a). The name “30” of this rule comes from the binary to decimal transformation of the string 00011110 obtained from the particular mapping of the eight configurations listed in the order shown in Fig. 1(a). ECA possesses two important symmetries: complementary and mirror. Here complementary means flipping the state in every site of the array. Hence, if Rule A and Rule B are complementary (mirror) symmetric, then running Rule A with a certain initial string will give the complementary (mirror) image of running Rule B with the complementary (mirror) version of that string. When these symmetries are taken into account, the number of unique rules reduces to 88 (and not 64 since mirror and/or complementary symmetries of certain rules are equivalent to themselves). Simple observations on ECA runs reflect visual structures Figure 1: (a) Representation of the Rule 30 in terms of operators. (b) Trans- of uniform, stable, oscillatory or irregular patterns. These struc- formations needed to switch between mirror and complementary symmetries of a rule. (c) Switching between the Rule 30 and its symmetries using operator tures are prone to a mixture of three types of fundamental iter- representation. t t+1 ations [si → si ], namely: decay [0|1 → 0], stability [0(1) → 0(1)], and growth [0|1 → 1]. Note that this approach becomes evident in a numerical representation of the state space, and it is under mirror transformation. Omitting one of these pairs erases the key in translating the rules into discrete operators. We first a whole column of repetitions. Continuing in this fashion one use the mirror symmetry of these systems to regroup the eight can reach at a 10×10 table that has all 88 unique Rules with possible configurations into symmetric and asymmetric sets, as only 12 repetitions. While constructing this table, one needs to seen in Fig. 1(a). Then, within each set, we group complemen- decide which repeating columns to erase and how to arrange the tary configurations together, leading to four groups (denoted by rows and the columns that are left at the end. The table that we Roman numerals in four different colors). Each group has cen- have constructed, after evaluating numerous options based on tral cells with values 0 and 1 that are mapped in four different mathematical and aesthetic criteria, is presented in Fig. 2. The ways: [0 → 0, 1 → 0], [0 → 0, 1 → 1], [0 → 1, 1 → 0], and 12 repetitions that appear at the corners of the table are removed [0 → 1, 1 → 1]. We call these double mappings operators, and for clarity. Note that, every adjacent row and column share at conveniently name them as Decay (D), Stability (S), Oscilla- least one common operator which means that every adjacent tion (O), and Growth (G), respectively. Note that the oscillation rule on the Table share at least three common operators. operator is a compound of decay and growth iterations. Four The Periodic Table presented in Fig. 2 offers a systematic groups and four possible operators cover all 256 rules (44 = 28). “bird’s eye” view of all 88 unique rules of ECA. Rules domi- The operator representation of Rule 30 becomes DSOG. Sym- nated by similar simple patterns (homogeneous, vertical lines, metric counterparts of rules are easily constructed in operator diagonal lines, horizontal stripes) tend to appear together. The representation. As shown in Fig. 1(b), to get the mirror symme- rules that show rich behavior populate the “fertile crescent” try of a rule, one needs to switch the operators in the group III along the diagonal where simple rules with contradicting pat- and group IV. To get the complementary of a rule, one needs to terns are expected to overlap. Among these rich rules, the ones replace all D operations (if any) with G and vice versa. Sym- that have common features are also brought together. Rule pairs metries of the Rule 30 (DSOG) found by these transformations 18, 146 and 122, 126 are striking examples of this. Despite the are presented as an example in Fig. 1(c). chaotic nature of these rules, starting a run with one of them Symmetric (I and II) and asymmetric (III and IV) sets of and switching to the other rule results in the same pattern that is operators are decoupled from each other with respect to both produced without the switching. This is because, Rule 18 (122) mirror and complementary transformations. Hence, it is in- and Rule 146 (126) share the same mapping, except for the con- structive to arrange ECA in a “periodic table” by placing possi- figuration 111 (010) which is mapped to 0 in the former and 1 ble symmetric sets as abscissa and asymmetric sets as ordinate. in the latter. This 111 (010) configuration is “washed out” in a However, using all 16 pairs of operations in both axes leads few steps and is never visited again. This effect is also present to many repetitions of rules that are identical under mirror and if one starts with the Rule 26 and continues with the Rule 154 complementary transformations. This can be avoided by real- but not the other way around. izing that, for example, a symmetric set “DO” becomes “GO” The Periodic Table of ECA also resonates with the findings under complementary transformation while remaining the same of Li and Packard [9] in their classic study on the structure of 2 Figure 2: Periodic table of the elementary cellular automata (ECA). Rules corresponding to operators representations (in the order I, II, III, IV) and their mirror and complementary counterparts (if different) are presented below each box in increasing order. Each box presents a run starting with a random sequence of 100 binary digits evolved for 100 time steps according to the Rule that is named by the smallest number. Periodic boundary condition is used. Chaotic, locally chaotic and complex rules are highlighted with red, blue and purple squares, respectively. Rules that acquire aperiodic behavior upon the logistic extension are highlighted with green squares.

3 outer-totalistic CA: Game of Life and Rule 90. This extension is achieved via introduction of a parameter, λ, that tunes the dynamics of CA. λ = 1 corresponds to the original binary version of the studied systems. As λ is tuned below 1, the binary state space extends into a Cantor set and the systems expand their complexity through series of deterministic transitions [14]. In particular, the Rule 90 which is aperiodic at λ = 1 shows complex (or Class 4) behavior at λ ∼ 0.6. The operator representation presented here enables us to go beyond the outer-totalistic rules and generalize the logistic extension to all ECA. We first define four regions of operation for each group (I, II, III and IV) as shown in Fig. 3. The coordinates of a configuration [L, C, R] (denoting left, center and right sites, respectively) defined as the 1 1 sums x ≡ L + R + 2 (mod 2) and y ≡ L + C + 2 (mod 2) de- termine in which operation region it falls. As shown in Fig. 3, the eight possible binary configurations appear at the centers of the regions that correspond to their group definitions shown in Fig 1(a). Hence, the configuration [L, C, R] determines the operation region which in turn determines the corresponding operator based on the rule at hand. Depending on the operator, the value of a site is updated according to the following formulae:

Figure 3: Definition of the operation regions based on a configuration. L, C and R correspond to the values at the left, center and right cell of a configuration. Decay ⇒ st+1 = (1 − λ)st Stability ⇒ st+1 = st the ECA rule space. They found two clusters of chaotic rules (in  (1 − λ)st + λ, if st ≤ 1 this context it includes the complex rules 54 and 110). Chaotic  2 Oscillation ⇒ st+1 =  A includes Rules 18, 22, 30, 54, 146, and 150 while Chaotic B   t t 1 has Rules 60, 90, 106, 110, 122, and 126. As seen in Fig 2, they (1 − λ)s , if s > 2 appear as clusters at the bottom left and top right of the “fertile Growth ⇒ st+1 = (1 − λ)st + λ crescent”, respectively. The authors found Rule 45 to be sepa- rated from the clusters but in our Table we find it connected to where st and st+1 are the values of the central site at the the cluster B. Furthermore, clusters A and B are connected over current and the next time step, respectively. These equations, a bridge of locally chaotic Rule 26 in the Table. There are no consistent with the operator notation, make up the new form of other chaotic rules in the row and the column of the Rule 105, the transition function. Note that this generalization is consis- which was also found to be isolated by Li and Packard, but it tent with the special case of the Rule 90 that we have reported is connected to the cluster B over a bridge of locally chaotic earlier [14]. Rule 73. Significant changes in dynamics can occur when x or y passes The operator representation can further illuminate the stud- over from one region to another. This happens when the sum ies on the computational irreducibility of ECA. In particular, it L + C or L + R is equal to the critical thresholds 0.5 or 1.5. The is interesting to examine the rules that are detached from the values that L, C, and R can take is dictated by the λ-dependent coarse-graining network investigated by Israeli and Goldenfeld Cantor set. Hence, one can expect these changes at the values [15]. They have shown that, Rule 105 can be course grained by of λ that mark the equality of binary sums to the critical thresh- the Rule 150. In the operator representation, these rules appear olds. As λ is tuned below 1, the first time such a transition as OOSS and SSOO, respectively. Furthermore, both DGDG occurs is when 2λ = 1.5. After this point, some of the rules (Rule 60) and DDGG (Rule 90) can be coarse-grained by them- start behaving differently than their original version. selves. Finally, the authors were unable to coarse-grain four Rules that exhibit chaotic, locally chaotic or complex be- unique rules: 30, 45, 106 and 154. In the operator represen- havior pass through multiple phase transitions while going be- tation, they happen to be DSOG, OGDS, OSGD, and SDOG. tween these regimes. As seen in Fig. 4, chaotic Rule 18 be- These make up four unique rules that involve all four opera- comes complex at λ = 0.73 mimicking (but not exactly copy- tors while avoiding two complementary symmetric operators ing) the complex patterns seen in one of its neighbors, Rule 54. (D and G) in the same mirror symmetric set. In other words, Another locally chaotic rule close by, Rule 82, also mimics the the rules that were found to be irreducible are the ones that ap- Rule 54 behavior at λ = 0.74. pear the most asymmetric in the operator representation. We Logistic extension breaks the symmetry between mirror rules believe that these mere observations can guide further studies because of the left-right asymmetry in the sum L + C. This is in this subject. clear in the distinct behavior of Rule 26 (the mirror symmetry Recently, we have introduced the logistic extension of two of Rule 82) which has a mixture of chaotic and locally chaotic 4 In summary, to understand the disjunctive and connective (diverse and unifying) nature of ECA rules, we redefine the transition function by introducing an operator-based notation. This allows one to organize the rules in a periodic table where underlying connections between their macroscopic behaviors and genetic codes emerge. Furthermore, we introduce a tuning parameter which controls the rate of rule iterations. This parameter extends the range of behavior that ECA can offer while generating inter-class transitions and disclosing inert behaviors of periodic rules. We believe that logistic extension to the operator-based representation may be useful to explore hid- den features in other complex systems, such as discrete lattice models [16] and boolean genetic networks [17]. S. J. acknowledges support from the Turkish Academy of Sciences - Outstanding Young Scientists Award Program (TUBA-¨ GEBIP).˙

References

[1] J. von Neumann, Theory of Self-Reproducing Automata, edited and com- pleted by A.W. Burks (University of Illinois Press, Urbana, IL, 1966). [2] S. Wolfram. Cellular automata as models of complexity, Nature 311 (1984) 419. [3] A. Deutsch and S. Dormann, Cellular Automaton Modeling of Biological Pattern Formation (Birkhauser,¨ Boston, 2005). [4] B. Chopard, in Computational Complexity: Theory, Techniques, and Ap- plications, edited by R. Meyers (Springer, New York, 2012), pp. 407–433. [5] S. Wolfram. Statistical mechanics of cellular automata, Rev. Mod. Phys. 55 (1983) 601. [6] L. Manukyan, S. A. Montandon, A. Fofonjka, S. Smirnov, and M. C. Milinkovitch. A living mesoscopic cellular automaton made of skin scales, Nature 544 (2017) 173. [7] S. Wolfram, A New Kind of Science, (Wolfram Media, Champaign, IL, Figure 4: 150×150 cells snapshots at a later stage of a 1000×1000 simulation 2002). for various Rules with given values of λ. The color bar shown at the top maps [8] C. Langton. Computation at the edge of chaos: Phase transitions and the range between the minimum and maximum cell values for each snapshot. emergent computation, Physica D 42 (1990) 12. Both conventional and operator representation of the rules are given below each [9] W. Li and N.H. Packard. The structure of the elementary cellular automata panel. rule space, Complex Systems 4 (1990) 281. [10] A. Shreim, P. Grassberger, W. Nadler, B. Samuelsson, J.E.S. Socolar, and M. Paczuski. Network analysis of the state space of discrete dynamical systems, Phys. Rev. Lett. 98 (2007) 198701. behavior at λ = 0.74. However, complementary rules behave [11] H.A. Gutowitz. A hierarchical classification of cellular automata, Physica in the same way under the logistic extension. For example, D 45 (1990) 136. the behavior of complementary Rules 90 and 165 are the same [12] P.M. Binder. A phase diagram for elementary cellular automata, Complex at λ = 0.6 [14]. Rules that originally have complex behavior Systems 7 (1993) 241. [13] R. Badii and A. Politi. Thermodynamics and Complexity of Cellular Au- may remain complex while having noticeable changes in their tomata, Phys. Rev. Lett. 78 (1997) 444. dynamics, for example Rule 54 at λ = 0.74. They also can [14] M. Ibrahimi, O. Gulseren,¨ S. Jahangirov. Deterministic phase transitions become locally chaotic like Rule 110 at λ = 0.74 or become and self-organization in logistic cellular automata, Phys. Rev. E 100 chaotic like Rule 124 (not shown in Fig. 4) which resembles its (2019) 042216. [15] N. Israeli and N. Goldenfeld. Computational Irreducibility and the Pre- neighboring chaotic Rule 60 at λ = 0.72. Rules that are chaotic dictability of Complex Physical Systems, Phys. Rev. Lett. 92 (2004) or locally chaotic can behave in a complex fashion as exempli- 074105. fied by Rule 86 (mirror symmetry of the Rule 30) at λ = 0.68 [16] K. Kaneko. Overview of coupled map lattices, Chaos 2 (1992) 279. λ . [17] S. A. Kauffman, The Origins of Order: Self-Organization and Selection and Rule 154 at = 0 68, respectively. in Evolution (Oxford University Press, New York, 1993). Some of the Rules that are originally periodic can acquire aperiodic behavior. For example, Rule 38 becomes locally chaotic at λ ∼ 0.69 and chaotic at λ ∼ 0.61. Periodic rules can also become complex, for example Rule 37 and Rule 46 at λ = 0.72, as shown in Fig. 4. Rules that gain aperiodic behavior upon the logistic extension are highlighted by the green squares in Fig. 2. Note that, these rules are adjacent to the rules that are originally aperiodic.